This paper investigates blocking probabilities
obtained from multidimensional truncated Poisson distributions.
For blocking probabilities typically arising in layered
cellular mobile communications networks, the large deviations
results of Gazdzicki et al. [9] are extended
to state spaces determined by multiple constraints. The
results yield asymptotically exact expressions that provide
an accurate approximation of probabilities up to 1%, which
considerably extends the applicability of large deviations
results and enables efficient approximation of blocking
probabilities for realistic mobile communications networks.